Optimal. Leaf size=25 \[ \frac {1}{8} \sinh ^{-1}\left (x^4\right )+\frac {1}{8} \sqrt {x^8+1} x^4 \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {275, 195, 215} \[ \frac {1}{8} \sqrt {x^8+1} x^4+\frac {1}{8} \sinh ^{-1}\left (x^4\right ) \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 275
Rubi steps
\begin {align*} \int x^3 \sqrt {1+x^8} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \sqrt {1+x^2} \, dx,x,x^4\right )\\ &=\frac {1}{8} x^4 \sqrt {1+x^8}+\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^4\right )\\ &=\frac {1}{8} x^4 \sqrt {1+x^8}+\frac {1}{8} \sinh ^{-1}\left (x^4\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.88 \[ \frac {1}{8} \left (\sinh ^{-1}\left (x^4\right )+\sqrt {x^8+1} x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 29, normalized size = 1.16 \[ \frac {1}{8} \, \sqrt {x^{8} + 1} x^{4} - \frac {1}{8} \, \log \left (-x^{4} + \sqrt {x^{8} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 29, normalized size = 1.16 \[ \frac {1}{8} \, \sqrt {x^{8} + 1} x^{4} - \frac {1}{8} \, \log \left (-x^{4} + \sqrt {x^{8} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 20, normalized size = 0.80 \[ \frac {\sqrt {x^{8}+1}\, x^{4}}{8}+\frac {\arcsinh \left (x^{4}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.10, size = 58, normalized size = 2.32 \[ \frac {\sqrt {x^{8} + 1}}{8 \, x^{4} {\left (\frac {x^{8} + 1}{x^{8}} - 1\right )}} + \frac {1}{16} \, \log \left (\frac {\sqrt {x^{8} + 1}}{x^{4}} + 1\right ) - \frac {1}{16} \, \log \left (\frac {\sqrt {x^{8} + 1}}{x^{4}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int x^3\,\sqrt {x^8+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.57, size = 19, normalized size = 0.76 \[ \frac {x^{4} \sqrt {x^{8} + 1}}{8} + \frac {\operatorname {asinh}{\left (x^{4} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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